Reference this comment -
"And also of the two, Catherine Carey was the elder sibling instead of her brother. That would make her half sister and full cousin to Elizabeth I. Also that would make the execution of Lettice Dudley's son, a necessity since he does have Royal blood through his grandmother..."
The Earl of Essex's illegitimate royal blood (if Catherine was Henry VIII's daughter) would be largely irevelant seeing as he had numerous legitimate Royal Line of descent which would have been far more of a problem.
Here's his main ones from Edward III - though he has many more descents from Edward I.
Edward III - Edmund Duke of York - Richard of Conisburgh, 3rd Earl of Cambridge - Isabel Plantagenet Countess of Essex - William Bourchier Viscount Bourchier - Cecily Bourchier - Walter Devereux 1st Viscount Hereford - Sir Richard Devereux - Walter 1st Earl of Essex - Robert 2nd Earl of Essex.
Edward III - Thomas of Woodstock Duke of Gloucester - Anne of Gloucester -(by her first marriage) Humphrey Stafford 1st Duke of Buckingham - Henry Stafford 2nd Duke of Buckingham - Anne Stafford - Dorothy Hastings - Walter 1st Earl of Essex - Robert 2nd Earl of Essex
Edward III - Thomas of Woodstock Duke of Gloucester - Anne of Gloucester - (by her second marriage) Henry Bourchier Earl of Essex - William Bourchier Viscount Bourchier - Cecily Bourchier - Walter Devereux 1st Viscount Hereford - Sir Richard Devereux - Walter 1st Earl of Essex - Robert 2nd Earl of Essex.
Edward III - Lionel Duke of Clarence - Phillippa Mortimer Countess of Ulster - Roger Mortimer 4th Earl of March - Anne Mortimer - Isabel Plantagenet Countess of Essex - William Bourchier Viscount Bourchier - Cecily Bourchier - Walter Devereux 1st Viscount Hereford - Sir Richard Devereux - Walter 1st Earl of Essex - Robert 2nd Earl of Essex.
Edward III - John of Gaunt Duke of Lancaster - Joan Beaufort - Anne Neville - Henry Stafford 2nd Duke of Buckingham - Anne Stafford - Dorothy Hastings - Walter 1st Earl of Essex - Robert 2nd Earl of Essex.
sorry bit off topic